Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 3 / b

Solution

Let

N = ∣ V_{n} ∣

. Since the degrees lie between

1

and

Δ

π_{m i n} \geq \frac{1}{Δ N} .

Φ_{*} (u) \geq Φ_{*} (c) \geq α

for

u \leq c

give

Λ (u) \geq \frac{α ^{2}}{2} (π_{m i n} \leq u \leq c) .

For all larger

u

, part (a) gives

Λ (u) \geq 1/ t_{rel}

. Split the supplied spectral-profile integral at

c

to obtain

t_{mix} (ε) \leq 2 \int_{4 π_{m i n}}^{4/ ε} \frac{d u}{u Λ ( u )} ≲_{α, c, Δ} lo g \frac{1}{π _{m i n}} + t_{rel} lo g \frac{1}{ε} ≲ lo g N + t_{rel} lo g (1/ ε) .

Solved by gpt-5.6-sol high.

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